Extension of the Darcy–Forchheimer Law for Shear-Thinning Fluids and Validation via Pore-Scale Flow Simulations

Abstract

Flow of non-Newtonian fluids through porous media at high Reynolds numbers is often encountered in chemical, pharmaceutical and food as well as petroleum and groundwater engineering and in many other industrial applications. Under the majority of operating conditions typically explored, the dependence of pressure drops on flow rate is non-linear and the development of models capable of describing accurately this dependence, in conjunction with non trivial rheological behaviours, is of paramount importance. In this work pore-scale single-phase flow simulations conducted on synthetic two-dimensional porous media are performed via computational fluid dynamics for both Newtonian and non-Newtonian fluids, and the results are used for the extension and validation of the Darcy-Forchheimer law, herein proposed for shear thinning fluid models of Cross, Ellis and Carreau. The inertial parameter β is demonstrated to be independent of the viscous properties of the fluids. The results of flow simulations show the superposition of two contributions to pressure drops: one, strictly related to the non-Newtonian properties of the fluid, dominates at low Reynolds numbers, while a quadratic one, arising at higher Reynolds numbers, is dependent on the porous medium properties. The use of pore-scale flow simulations on limited portions of the porous medium is here proposed for the determination of the macroscale averaged parameters (permeability K, inertial coefficient β and shift factor α), which are required for the estimation of pressure drops via the extended Darcy-Forchheimer law. The method can be applied for those fluids which would lead to critical conditions (high pressures for low permeability media and/or high flow rates) in laboratory tests.

Pore-scale simulation of fluid flow and solute dispersion in three-dimensional porous media

“…(5) are neglected and the momentum balance equation is applied to a simple one-dimensional problem, the original phenomenological (linear) law developed by Darcy is retrieved. At higher superficial velocities, the Darcy-Forchheimer law is generally used [5][6][7], and the friction coefficient is assumed to be function of the fluid superficial velocity: γ = μ /ρα + β|V|, where now α and β are the two Darcy-Forchheimer parameters. The application of the Darcy-Forchheimer law to a simple one-dimensional problem, again neglecting the time derivative and the inertial terms, results in a quadratic law, often written in the following form:…”

Non-Darcy displacement by a non-Newtonian fluid in porous media according to the Barree-Conway model

“…However, the recent research results (Tosco et al, 2013) indicate that the inertial effect and coefficients are demonstrated to be independent of the viscous properties of the fluids, and the above Equation (10) may be directly used in Equation (2).…”

Shear Rate Determination from Pore-Scale Flow Fields

“…To give an example, surfactant flooding targets the residual oil by enhancing the microscopic displacement efficiency. Polymer flooding on the other hand mainly targets the bypassed oil by improving the macroscopic sweep efficiency (Reuvers and Golombok 2009;van der Plas and Golombok 2015). There is also an effect on the microscopic displacement efficiency which can have also significant economic benefit.…”

“…Depending on the polymer formulation, these effects sometimes play a significant role in controlling the mobility reduction factor (Sorbie 1991;Rodriguez et al 2014). The increase in in situ viscosity is a combination of rheological effects such as shear-dependent fluid viscosity, extensional viscosity (Koroteev et al 2013;van der Plas and Golombok 2015) and visco-elasticity (Wang et al 2010). Polymer in general and more specifically aqueous solutions of hydrosoluble polymer including HPAM exhibit a shear rate-dependent bulk viscosity (Delshad et al 2008).…”

“…A direct link between the bulk viscosity μ = μ(γ ) and Darcy scale μ eff = μ eff (v Darcy ) can be made by using approximate analytical models (Fayed et al 2016) or hydrodynamic simulation of the pore-scale flow field for the non-Newtonian rheology (Balhoff 2000;Afshar-poor et al 2012;Clemens et al 2012;Afsharpoor and Balhoff 2013;Tosco et al 2013). Even though these approaches become more and more available, the most commonly used approach are semi-empirical correlations for the Darcy-scale effective shear rate (Delshad et al 2008).…”